What Function Types and Inverses Can Exist Between Different Cardinalities?

[ded8381]

Function Types: continuous, surjective, injective, bijective, continuous surjective, continuous injective, continuous bijective. Then all of the above -- with each possible type of inverse?

What is possible with F:X-->Y where X, Y can be [0,1], (0,1), [0,1), Q, R, N?

I certainly don't expect a full listed answer for each combination, but some general principles would be great. :)

I already know there couldn't be bijections between sets of different cardinality. And I know there couldn't be an injection from greater to lower, or a surjection from smaller to greater cardinality.

Thanks

David

 

[arkajad]

Specifying, for instance [0,1), is not enough. You need to specify the topology. The same set can be equipped with different topologies.

 

[VeeEight]

I'm assuming he means the standard Euclidean topology

 

[arkajad]

Well, then maps can be classified, for instance, by their differentiability properties - whenever applicable, and there are infinitely many classes. But why would one ask such question?

 

[ded8381]

Yes the Euclidean metric. I'm not really interested in differentiability types right now -- just the ones listed (surjective, injective, etc...) Maybe this is the wrong folder to ask -- but the question came to me while studying topology so it seemed appropriate.

David